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annotate scripts/special-matrix/hadamard.m @ 11587:c792872f8942
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| author | John W. Eaton <jwe@octave.org> |
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| date | Thu, 20 Jan 2011 17:35:29 -0500 |
| parents | fd0a3ac60b0e |
| children | 4d777e05d47c |
| rev | line source |
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| 11523 | 1 ## Copyright (C) 1993-2011 Paul Kienzle |
| 5827 | 2 ## |
| 3 ## This file is part of Octave. | |
| 4 ## | |
| 5 ## Octave is free software; you can redistribute it and/or modify it | |
| 6 ## under the terms of the GNU General Public License as published by | |
| 7016 | 7 ## the Free Software Foundation; either version 3 of the License, or (at |
| 8 ## your option) any later version. | |
| 5827 | 9 ## |
| 10 ## Octave is distributed in the hope that it will be useful, but | |
| 11 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 13 ## General Public License for more details. | |
| 14 ## | |
| 15 ## You should have received a copy of the GNU General Public License | |
| 7016 | 16 ## along with Octave; see the file COPYING. If not, see |
| 17 ## <http://www.gnu.org/licenses/>. | |
| 5827 | 18 ## |
| 19 ## Original version by Paul Kienzle distributed as free software in the | |
| 20 ## public domain. | |
| 21 | |
| 22 ## -*- texinfo -*- | |
| 23 ## @deftypefn {Function File} {} hadamard (@var{n}) | |
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24 ## Construct a Hadamard matrix @var{Hn} of size @var{n}-by-@var{n}. The |
| 6516 | 25 ## size @var{n} must be of the form @code{2 ^ @var{k} * @var{p}} in which |
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26 ## @var{p} is one of 1, 12, 20 or 28. The returned matrix is normalized, |
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27 ## meaning @code{Hn(:,1) == 1} and @code{Hn(1,:) == 1}. |
| 5827 | 28 ## |
| 29 ## Some of the properties of Hadamard matrices are: | |
| 30 ## | |
| 31 ## @itemize @bullet | |
| 32 ## @item | |
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33 ## @code{kron (@var{Hm}, @var{Hn})} is a Hadamard matrix of size |
| 5827 | 34 ## @var{m}-by-@var{n}. |
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35 ## |
| 5827 | 36 ## @item |
| 6248 | 37 ## @code{Hn * Hn' == @var{n} * eye (@var{n})}. |
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38 ## |
| 5827 | 39 ## @item |
| 40 ## The rows of @var{Hn} are orthogonal. | |
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41 ## |
| 5827 | 42 ## @item |
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43 ## @code{det (@var{A}) <= abs(det (@var{Hn}))} for all @var{A} with |
| 5827 | 44 ## @code{abs (@var{A} (@var{i}, @var{j})) <= 1}. |
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45 ## |
| 5827 | 46 ## @item |
| 47 ## Multiply any row or column by -1 and still have a Hadamard matrix. | |
| 48 ## @end itemize | |
| 49 ## | |
| 50 ## @end deftypefn | |
| 51 | |
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52 |
| 5827 | 53 ## Reference [1] contains a list of Hadamard matrices up to n=256. |
| 54 ## See code for h28 in hadamard.m for an example of how to extend | |
| 55 ## this function for additional p. | |
| 56 ## | |
| 57 ## References: | |
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58 ## [1] A Library of Hadamard Matrices, N. J. A. Sloane |
| 5827 | 59 ## http://www.research.att.com/~njas/hadamard/ |
| 60 | |
| 61 function h = hadamard (n) | |
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62 |
| 5827 | 63 if (nargin != 1) |
| 64 print_usage (); | |
| 65 endif | |
| 66 | |
| 8507 | 67 ## Find k if n = 2^k*p. |
| 5827 | 68 k = 0; |
| 69 while (n > 1 && floor (n/2) == n/2) | |
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70 k++; |
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71 n = n/2; |
| 5827 | 72 endwhile |
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73 |
| 8507 | 74 ## Find base hadamard. |
| 75 ## Except for n=2^k, need a multiple of 4. | |
| 5827 | 76 if (n != 1) |
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77 k -= 2; |
| 5827 | 78 endif |
| 79 | |
| 8507 | 80 ## Trigger error if not a multiple of 4. |
| 5827 | 81 if (k < 0) |
| 82 n =- 1; | |
| 83 endif | |
| 84 | |
| 85 switch (n) | |
| 86 case 1 | |
| 87 h = 1; | |
| 88 case 3 | |
| 89 h = h12 (); | |
| 90 case 5 | |
| 91 h = h20 (); | |
| 92 case 7 | |
| 93 h = hnormalize (h28 ()); | |
| 94 otherwise | |
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95 error ("hadamard: N must be 2^k*p, for p = 1, 12, 20 or 28"); |
| 5827 | 96 endswitch |
| 97 | |
| 8507 | 98 ## Build H(2^k*n) from kron(H(2^k),H(n)). |
| 5827 | 99 h2 = [1,1;1,-1]; |
| 100 while (true) | |
| 101 if (floor (k/2) != k/2) | |
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102 h = kron (h2, h); |
| 5827 | 103 endif |
| 104 k = floor (k/2); | |
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105 if (k == 0) |
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106 break; |
| 5827 | 107 endif |
| 108 h2 = kron (h2, h2); | |
| 109 endwhile | |
| 110 endfunction | |
| 111 | |
| 112 function h = h12 () | |
| 8507 | 113 tu = [-1,+1,-1,+1,+1,+1,-1,-1,-1,+1,-1]; |
| 114 tl = [-1,-1,+1,-1,-1,-1,+1,+1,+1,-1,+1]; | |
| 115 ## Note: assert (tu(2:end), tl(end:-1:2)). | |
| 5827 | 116 h = ones (12); |
| 117 h(2:end,2:end) = toeplitz (tu, tl); | |
| 118 endfunction | |
| 119 | |
| 120 | |
| 121 function h = h20 () | |
| 8507 | 122 tu = [+1,-1,-1,+1,+1,+1,+1,-1,+1,-1,+1,-1,-1,-1,-1,+1,+1,-1,-1]; |
| 123 tl = [+1,-1,-1,+1,+1,-1,-1,-1,-1,+1,-1,+1,-1,+1,+1,+1,+1,-1,-1]; | |
| 124 ## Note: assert (tu(2:end), tl(end:-1:2)). | |
| 5827 | 125 h = ones (20); |
| 126 h(2:end,2:end) = fliplr (toeplitz (tu, tl)); | |
| 127 endfunction | |
| 128 | |
| 129 function h = hnormalize (h) | |
| 8507 | 130 ## Make sure each row and column starts with +1. |
| 5827 | 131 h(h(:,1)==-1,:) *= -1; |
| 132 h(:,h(1,:)==-1) *= -1; | |
| 133 endfunction | |
| 134 | |
| 135 function h = h28 () | |
| 136 ## Williamson matrix construction from | |
| 137 ## http://www.research.att.com/~njas/hadamard/had.28.will.txt | |
| 8507 | 138 s = ["+------++----++-+--+-+--++--"; |
| 139 "-+-----+++-----+-+--+-+--++-"; | |
| 140 "--+-----+++---+-+-+----+--++"; | |
| 141 "---+-----+++---+-+-+-+--+--+"; | |
| 142 "----+-----+++---+-+-+++--+--"; | |
| 143 "-----+-----++++--+-+--++--+-"; | |
| 144 "------++----++-+--+-+--++--+"; | |
| 145 "--++++-+-------++--+++-+--+-"; | |
| 146 "---++++-+-----+-++--+-+-+--+"; | |
| 147 "+---+++--+----++-++--+-+-+--"; | |
| 148 "++---++---+----++-++--+-+-+-"; | |
| 149 "+++---+----+----++-++--+-+-+"; | |
| 150 "++++--------+-+--++-++--+-+-"; | |
| 151 "-++++--------+++--++--+--+-+"; | |
| 152 "-+-++-++--++--+--------++++-"; | |
| 153 "+-+-++--+--++--+--------++++"; | |
| 154 "-+-+-++--+--++--+----+---+++"; | |
| 155 "+-+-+-++--+--+---+---++---++"; | |
| 156 "++-+-+-++--+------+--+++---+"; | |
| 157 "-++-+-+-++--+------+-++++---"; | |
| 158 "+-++-+---++--+------+-++++--"; | |
| 159 "-++--++-+-++-+++----++------"; | |
| 160 "+-++--++-+-++-+++-----+-----"; | |
| 161 "++-++---+-+-++-+++-----+----"; | |
| 162 "-++-++-+-+-+-+--+++-----+---"; | |
| 163 "--++-++++-+-+----+++-----+--"; | |
| 164 "+--++-+-++-+-+----+++-----+-"; | |
| 165 "++--++-+-++-+-+----++------+"]; | |
| 5901 | 166 ## Without this, the assignment of -1 will not work properly |
| 167 ## (compatibility forces LHS(idx) = ANY_VAL to keep the LHS logical | |
| 168 ## instead of widening to a type that can represent ANY_VAL). | |
| 8507 | 169 h = double (s == "+"); |
| 5827 | 170 h(!h) = -1; |
| 171 endfunction | |
| 172 | |
| 173 %!assert(hadamard(1),1) | |
| 174 %!assert(hadamard(2),[1,1;1,-1]) | |
| 175 %!test | |
| 176 %! for n=[1,2,4,8,12,24,48,20,28,2^9] | |
| 177 %! h=hadamard(n); assert(norm(h*h'-n*eye(n)),0); | |
| 178 %! end |